On the densest lattice packing of centrally symmetric octahedra

Abstract:

The main purpose of this paper is the calculation of the critical determinant, and therefore the packing constant, for any centrally symmetric octahedron. The results are obtained partially by a numerical computation that is not rigorous. As an application, we prove that the lattice of integer vectors perpendicular to any integer vector ${\mathbf {n}} = [{n_1},{n_2},{n_3},{n_4}]\;(0 \leq {n_1} \leq {n_2} \leq {n_3} \leq {n_4},{n_4} > 0)$ contains a nonzero vector ${\mathbf {m}} \in {{\mathbf {Z}}^4}$, the height $(h({\mathbf {m}}) = \max |{m_i}|)$ of which satisfies \[ \begin {array}{*{20}{c}} {{\text {(i)}}} \\ {{\text {(ii)}}} \\ {{\text {(iii)}}} \\ \end {array} \quad \begin {array}{*{20}{c}} {h({\mathbf {m}}) < {{(\tfrac {4}{3}h({\mathbf {n}}))}^{1/3}}\quad {\text {if}}\;{n_4} \leq - 2{n_1} + {n_2} + {n_3},} \hfill \\ {h({\mathbf {m}}) < {{(\tfrac {{27}}{{19}}h({\mathbf {n}}))}^{1/3}}\quad {\text {in any case,}}} \hfill \\ {h({\mathbf {m}}) \leq h{{({\mathbf {n}})}^{1/3}}\quad {\text {if}}\;{n_4} \geq {n_1} + {n_2} + {n_3}.} \hfill \\ \end {array} \] The closing examples show that the above estimations cannot be improved.